5 Pirates Solution is wrong - here is the real solution

mathdad · 2013-04-13 13:57:18

This is for the version where all pirates get to vote:

Forum Solution:
The oldest pirate will propose a 98 : 0 : 1 : 0 : 1 split, in other words the oldest pirate gets 98 coins, the middle pirate gets 1 coin and the youngest gets 1 coin.

Let us name the pirates (from oldest to youngest): Alex, Billy, Colin, Duncan and Eddie.

Working backwards:

2 Pirates: Duncan splits the coins 100 : 0 (giving himself all the gold). His vote (50%) is enough to ensure the deal.

3 Pirates: Colin splits the coins 99 : 0 : 1. Eddie will accept this deal (getting just 1 coin), because he knows that if he rejects the deal there will be only two pirates left, and he gets nothing.

- Incorrect. Eddie will not accept the deal. This is not about gold. Instead, it is about life and death. Who is really in control here? The premise is that Colin is in control at this point, and can be greedy by offering Eddie just a single gold coin. But, the reality is that Colins life is in jeopardy. If Colin doesnt please Eddie, he will die. Eddie knows this, and he will not settle for just 1 coin. Eddie is in complete control and will extort Colin to the fullest extent possible. In fact, since pirates are bloodthirsty and greedy, Colin had better offer Eddie all the gold, or Eddie will vote against his proposal and he will die. What good is all the gold in the world if you are dead? Thus, to save his own life, the only proposal he can make is to give all the gold away. And, since it is Colins proposal, Colin must vote in favor of it, because Duncan will definitely vote no, having been offered nothing. Eddie, now having 100% of the gold will also, finally, vote in favor of it. If Eddie votes no, then he will get nothing from Duncans proposal when only 2 pirates remain.

- Poor Colin is thinking hard about his dilemma. Maybe he can strike a deal with Duncan instead. Colins only recourse is to offer Duncan all the gold instead of Eddie. But, that also will not work. While Duncan would get 100% of the gold here, he will still vote against it in favor of killing off Colin. Once Colin is gone, he gets all the gold anyway due to the 2 Pirates remaining rule. Thus, Colin must offer all 100 coins to Eddie, and Colin must accept his own deal or he will die.

4 Pirates: Poor Billy is in the same boat. He will die unless someone besides him accepts his proposal. What can he do?

Billy cant offer it to Eddie, because Eddie will get the same proposal from Colin after Billy is gone. Eddie would vote against Billy in this case, even if Billy offered him all the gold, in favor of killing him off. Eddie would stand to get all the gold all the time with Billy gone.

That leaves Duncan and Colin.

Make no mistake. The pirates are greedy and bloodthirsty. Billy must offer all the gold to one of them of he will die. It doesnt matter which one he chooses. He must only please one of them, and accept his own proposal. Assuming that there will be future spoils to share, his best strategy would be to alternate between the two pirates each time there is gold to be split.

5 Pirates: Alex must have 3 votes in acceptance of his proposal or he will die. Obviously, he will vote for his own proposal, but can he get the other 2?

Duncan and Colin are better off with Alex gone (which would force Billy to select one of them to receive all the gold). Due to their thirst for blood and greed, they would opt for seeing Alex die and having a 1 in 2 chance of getting all the gold. They would not accept anything short of 100% of the gold from Alex. However, if Alex offers 100% of the gold to only 1 other pirate, he will only have 2 acceptance votes (his own, and the pirate receiving all 100 gold coins). Thus, Alex cannot make a deal with either of these two.

That leaves Billy and Eddie. Alex must give part of the gold to each, and have each of them accept the proposal. Is that possible?

Billy knows that if he demands 100% of the gold for himself, then he would get nothing because Eddie would not accept, and Alex would die. And, if Alex dies, then we have to drop to the 4 Pirates solution. In the four pirates solution, neither Billy nor Eddie gets anything.

From Eddies perspective, he also knows that he gets nothing from having Alex die. Billy is in no danger, and has a solution for staying alive in the four pirates solution.

Both Billy and Eddie have the power to accept or reject Alexs proposal. They both need the other to accept the proposal to get paid. If either of them rejects the proposal, then neither of them will get anything. Their positions are equal, and Alex has only one thing that he can do. Offer each of Billy and Eddie 50 gold coins.

Thus, the final solution is 0-50-0-0-50.

anonimnystefy · 2013-04-13 14:04:03

Let's talk about the 3 case first, because that is where the disagreement begins.

If you were Eddie and I was Collin, and I proposed to you the 99:0:1 deal, would you accept (knowing that you will get nothing if I die)?

mathdad · 2013-04-13 15:13:10

anonimnystefy wrote:

Let's talk about the 3 case first, because that is where the disagreement begins.
If you were Eddie and I was Collin, and I proposed to you the 99:0:1 deal, would you accept (knowing that you will get nothing if I die)?

No I wouldn't, because as Eddie, I know I am in control and can do much better for myself. Let me pose this question to you:

You a making a proposal to me and to one other person. You know full well that the other person will kill you regardless of what you propose. Knowing full well that you are going to die if you don't make me happy, would you still propose to offer me only 1 coin, keeping 99 for yourself? Realize of course that you only have one opportunity to make me happy before you die. Also realize that I am a bloodthirsty,greedy pirate who kills and steals for a living. And, that for 1 measly coin, I would probably rather see you die for your insulting offer? And, realize that you can't take the gold with you if you are dead. Your life is entirely in my hands, and you propose that I should only get 1 coin and you get to keep 99 for sparing your life?

anonimnystefy · 2013-04-13 15:17:22

I think you are getting the pirate setup too seriously. The point of the problem is that each pirate wants to maximize the amount of money he gains.

Nehushtan · 2013-04-13 15:20:53

mathdad wrote:

anonimnystefy wrote:
Let's talk about the 3 case first, because that is where the disagreement begins.
If you were Eddie and I was Collin, and I proposed to you the 99:0:1 deal, would you accept (knowing that you will get nothing if I die)?
No I wouldn't, because as Eddie, I know I am in control and can do much better for myself.

No, Eddie most certainly CANNOT do better. Under Colins proposal Eddie will get 1 gold. If Eddie does not accept, then he will end up with nothing because Colin will die and Duncan will propose to keep all the gold for himself. No matter what Eddie does, there is no way he can get more than 1 gold. Since 1 gold is better than no gold, Eddie is forced to vote for Colins split.

Last edited by Nehushtan (2013-04-13 15:29:39)

mathdad · 2013-04-13 15:38:58

The parameters of the puzzle are not only to maximize your profits, but to stay alive as well.

What you need to consider (as Colin) are your possible outcomes:

1. you die, and Duncan gets all the gold.

2. you live, and Eddie gets all the gold.

Either way, you get no gold. Your only decision here is whether you want to live or die.

Put yourself into the problem, and pretend it is real. What would you really do?

I can guarantee that nobody would propose offering Eddie 1 coin.

Would you die just to keep me from getting gold? You can't take it with you.

Seriously? You wouldn't even consider offering Eddie even 2 coins? 10?

The reality is that you would offer Eddie all the gold, your sword, and the clothes off your back.

I didn't write the puzzle. I am only operating within what is written.

anonimnystefy · 2013-04-13 15:42:40

You are incorrectly assuming that Eddie would rather see Collin dead than get 1 coin.

mathdad · 2013-04-13 15:46:37

Not true, the parameters of the puzzle are to maximize your profits and stay alive.

But, this is all moot anyway, because case 3 never ever comes into play.

No pirates die, and Alex offers both Billy and Eddie each 50 gold coins. They accept the offer.

That is the solution.

anonimnystefy · 2013-04-13 15:50:02

It is not.

We need to dicuss the 3 case first, otherwise there is no way we can come to an agreement on 5.

Let's say that Collin still proposes the 99:0:1 deal. What is Eddie supposed to do?

mathdad · 2013-04-13 16:01:19

anonimnystefy wrote:

It is not.
We need to dicuss the 3 case first, otherwise there is no way we can come to an agreement on 5.
Let's say that Collin still proposes the 99:0:1 deal. What is Eddie supposed to do?

You have missed a step.

First: Colin must maximize his profits AND stay alive. He is not psychic and can't say what any other pirate will do. He must guarantee that he stays alive as his first priority. Secondary, he must try to gain gold for himself.

Second: Eddie must make a decision depending on Colin's proposal.

Thus, what proposal MUST Colin make?

Nehushtan · 2013-04-13 16:01:40

mathdad wrote:

The parameters of the puzzle are not only to maximize your profits, but to stay alive as well.

Exactly. Colin KNOWS that he will NOT die because he knows that Eddie will NOT vote against his proposal. That is why he gives Eddie 1 gold. (If Eddie does not like it and gets rid of Colin he will get nothing; therefore Eddie votes to keep Colin alive and pocket the 1 gold.)

anonimnystefy · 2013-04-13 16:04:15

But, look. Collin is considering his options. If he splits 99:0:1, Eddie will get either 1 and Collin stays alive or get 0 and Collin is dead. Since the first option is better for Eddie, he knows that Eddie will go after that. He proposes 99:0:1.

mathdad · 2013-04-13 16:18:18

Again, the puzzle starts with 5 pirates, not 3. Case 3 is irrelevant.

Case 5 does NOT depend on case 3. Case 5 comes first.

Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

Again, it is irrational to assume that a greedy, bloodthirsty pirate would accept 1 gold coin.
It is also irrational to assume that you would live with such a pittance.

And

"As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard."

This line here allows each pirate to calculate what will happen in future dealings.

And so, if you propose that Alex can offer 98 for himself, 1 for Collin, and 1 for Eddie (This is THE official solution posted on this site......), then tell me, by what is written in the previous quote "if a pirate would get the same number of coins if he voted for or against a proposal, he will vote AGAINST....." means?

It means that Eddie must vote AGAINST receiving 1 coin, because that is what you are offering in step 3 - just a single coin to Eddie.

This is NOT allowed by the puzzle, you MUST vote to not accept the official solution to the problem, and kill the pirate Alex.

Last edited by mathdad (2013-04-13 16:19:47)

anonimnystefy · 2013-04-13 16:20:43

Hm, you are avoiding talking about the 3 case. Does that mean you agree with me on it?

Nehushtan · 2013-04-13 16:28:35

mathdad wrote:

Again, it is irrational to assume that a greedy, bloodthirsty pirate would accept 1 gold coin.
It is also irrational to assume that you would live with such a pittance.

This is NOT stated in the puzzle. This is something you made up yourself. When you play a game, you follow the rules of the game. You DO NOT make up your own rules!

Okay, look at it this way. Colin reasons in the following line.

1. Suppose he keeps all gold and gives nothing to Duncan and Eddie. Then both will vote no and he will die. So this is no good.

2. Suppose he takes 99 gold and gives 1 gold to one of the the other two. Should he give to Duncan or Eddie? Suppose Duncan. Duncan will vote no, obviously. How will Eddie vote? Eddie gets nothing here. If Colin dies and Duncan proposes, he will get nothing too. Either way he will get nothing. Therefore he will vote no because of the following rule:

Game rule wrote:

if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against

So this is no good to Colin either.

3. Suppose Colin takes 99 gold and gives 1 gold to Eddie. Duncan will vote no as before, but what matters is Eddies vote. If Colin dies Eddie will get no gold so Eddie has to agree to Colins proposal and accept 1 gold rather than no gold. So this works and is the scenario for the three-pirate case.

mathdad wrote:

Again, the puzzle starts with 5 pirates, not 3. Case 3 is irrelevant.
Case 5 does NOT depend on case 3. Case 5 comes first.

In order to understand the five-pirate case, you have to understand the three-pirate case first.

Last edited by Nehushtan (2013-04-13 16:30:20)

mathdad · 2013-04-13 16:30:50

Is has been my contention from the start that case 3 doesn't matter, because the puzzle starts and ends at case 5.

But, if we consider case 3, Eddie must not accept 1 coin at case 5.

That will take us to step 4, or another solution at case 5 exists.

Either way, the official solutions is still in question.

Step 3 depends on what happens in case 5, and if it goes past that, case 4., .....

I don't agree that it is rational or intelligent to believe that an offer of 1 coin is suitable for guaranteeing that you will stay alive. Again, parameters of the puzzle.

Nehushtan · 2013-04-13 16:41:04

mathdad wrote:

Is has been my contention from the start that case 3 doesn't matter, because the puzzle starts and ends at case 5.

As I said, you have to understand the three-pirate case in order to figure out the five-pirate case.

mathdad wrote:

I don't agree that it is rational or intelligent to believe that an offer of 1 coin is suitable for guaranteeing that you will stay alive.

That has nothing to do with the rules of the puzzle. Its just an opinion.

mathdad wrote:

Again, parameters of the puzzle.

Which you dont seem to be able to understand.

Last edited by Nehushtan (2013-04-13 17:17:49)

mathdad · 2013-04-13 16:46:04

Nehushtan wrote:

mathdad wrote:
Again, it is irrational to assume that a greedy, bloodthirsty pirate would accept 1 gold coin.
It is also irrational to assume that you would live with such a pittance.
This is NOT stated in the puzzle. This is something you made up yourself. When you play a game, you follow the rules of the game. You DO NOT make up your own rules!

Here is the URL of the puzzle:
OK, URL not allowed.....

w-w-w-.-m-a-t-h-I-s-f-u-n-.-c-o-m-/-p-u-z-z-l-e-s-/-5---p-I-r-a-t-e-s-.-h-t-m-l

Again, it says that the pirates are rational and intelligent. No specific definitions of those words were given. We all know that in real life, it would be completely irrational to offer 1 gold coin in that position and expect a greedy, bloodthirsty pirate to accept the offer.

mathdad · 2013-04-13 16:57:39

And, the ONLY rational, intelligent thing to do in that position would be to fight for your life, and make the absolute best offer possible. And, that would be for Collin to offer Eddie 100 gold, and save his life. Making intelligent, rational decisions about saving your life is a parameter of the puzzle. I believe I understand the parameters. What would any rational, intelligent person do in this situation?

Again, we are still debating - there is no need to start the name calling. Keep the debate on the puzzle.

anonimnystefy · 2013-04-13 17:00:15

There is no namecalling.

You should understand that the rules of the problem must be followed and not your opinion.

Nehushtan · 2013-04-13 17:03:06

mathdad wrote:

Again, it says that the pirates are rational and intelligent. No specific definitions of those words were given. We all know that in real life, it would be completely irrational to offer 1 gold coin in that position and expect a greedy, bloodthirsty pirate to accept the offer.

Why is it irrational IN A PUZZLE to have a pirate accept 1 gold because otherwise he would have nothing at all? If this situation offends your sensibility, you can always assume that later on, the pirate having the most gold gets murdered in his sleep, the other pirates start squabbling over the gold, etc. Do what you like with your imagination outside the puzzle. However, when you are in the puzzle itself, you go by the rules of the puzzle. You dont go making up your own.

Last edited by Nehushtan (2013-04-13 17:13:22)

mathdad · 2013-04-13 17:20:02

anonimnystefy wrote:

There is no namecalling.
You should understand that the rules of the problem must be followed and not your opinion.

I wasn't referring to you, but the other guy who said that I didn't understand and was making stuff up.

mathdad · 2013-04-13 17:30:38

Nehushtan wrote:

mathdad wrote:
Again, it says that the pirates are rational and intelligent. No specific definitions of those words were given. We all know that in real life, it would be completely irrational to offer 1 gold coin in that position and expect a greedy, bloodthirsty pirate to accept the offer.
Why is it irrational IN A PUZZLE to have a pirate accept 1 gold because otherwise he would have nothing at all? If this situation offends your sensibility, you can always assume that later on, the pirate having the most gold gets murdered in his sleep, the other pirates start squabbling over the gold, etc. Do what you like with your imagination outside the puzzle. However, when you are in the puzzle itself, you go by the rules of the puzzle. You dont go making up your own.

And likewise, why is it rational? Why does the definition of the word "rational" change because we are in the context of a puzzle? The puzzle set the stage, and the pirates are described as bloodthirsty and greedy. Are you saying that Eddie is greedy by accepting 1 gold coin? Is that greedy? Who would agree that this is greedy? Colin is being greedy. What allows for Collin to be more greedy than Eddie in this puzzle? Answer that.

And, tell me, by the parameters in the puzzle, how you can say that Eddie's "greed" of accepting 1 gold coin will outweigh his bloodthirstiness? The puzzle doesn't define those limits, or what weighs more than the other. You can't assume that the bloodthirstiness will be appeased by 1 gold coin. The maximum limit we have here is 100 gold coins, and if it is at all possible in the puzzle to appease the bloodthirstiness, then we have no other data, other than 100 Gold to go with. Thus, if Collin wants to appease Eddie's bloodthirstiness, he must offer his maximum to appease it. That maximum is 100 Gold.

anonimnystefy · 2013-04-13 17:31:14

I don't know if I would call the things he said namecalling, but they do seem a bit harsh. I am sure it was unintentional.

Nehushtan · 2013-04-13 17:46:24

mathdad wrote:

And likewise, why is it rational? Why does the definition of the word "rational" change because we are in the context of a puzzle?

It is rational because accepting 1 gold is better than accepting none.

mathdad wrote:

Are you saying that Eddie is greedy by accepting 1 gold coin? Is that greedy?

Yes. Why shouldnt it be?

However, what is more relevant to the puzzle is Colins greed. Colin would like to keep all 100 gold to himself. But he cant hell be outvoted by Duncan and Eddie. His next best option is to keep 99 and give Eddie 1 gold. And Eddie would prefer 1 gold to no gold because hes greedy.

mathdad wrote:

And, tell me, by the parameters in the puzzle, how you can say that Eddie's "greed" of accepting 1 gold coin will outweigh his bloodthirstiness?

Read this rule again:

As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.

The pirates are bloodthirsty only in so far as their greed is not satisfied. If by being bloodthirsty they get less instead of more (as would happen to Eddie if he voted against Colins proposal) then they wont be bloodthirsty, preferring to be greedy instead.

Last edited by Nehushtan (2013-04-13 18:23:44)

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#1 2013-04-13 13:57:18

5 Pirates Solution is wrong - here is the real solution

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Re: 5 Pirates Solution is wrong - here is the real solution

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Re: 5 Pirates Solution is wrong - here is the real solution

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